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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Calculate y'.

39. y = tan2(sin θ)

To determine

To calculate:  The derivative of y.

Explanation

Given:

The function is y=tan2(sinθ).

Result used: Chain Rule

If g is differentiable at θ and f is differentiable at g(θ), then the composite function F=fg defined by F(θ)=f(g(θ)) is differentiable at θ and F is given by the product,

F(θ)=f(g(θ))g(θ) (1)

Derivative Rule:

Power Rul: ddx(xn)=nxn1

Calculation:

Obtain the derivative of y.

y=ddx(y)=ddx(tan2(sinθ))

Let h(θ)=tan(sinθ) and f(u)=u2  where u=h(θ).

Apply the chain rule as shown in equation (1),

y=f(h(θ))h(θ) (2)

The derivative f(h(θ)) is computed as follows,

f(h(θ))=f(u)=ddu(f(u))=ddu(u2)

Apply the power rule then substitute u=tan(sinθ),

f(h(θ))=2u21=2u=2tan(sinθ)

Thus, the derivative is f(h(θ))=2tan(sinθ).

The derivative of h(θ) is computed as follows,

h(θ)=ddθ(tan(sinθ))

Let k(θ)=sinθ and g(u)=tanu  where u=k(θ)

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